Impact Cratering

Rocks and Shocks

The Hugoniot Elastic Limit

Rocks typically deform elastically by about 10-6 per bar of pressure, or about 10-11 per pascal. We say they have a Young's Modulus of about 100 Gpa (that is, 1/10-11).
Think of Young's Modulus as the stress it would take to deform rocks by 100%. This figure, of course, is higher for very strong rocks and much lower for weak rocks.
And of course the behavior of the rocks would change long before we reached 100%
deformation.

At low confining pressures, rocks typically can withstand a few kilobars (a few hundred Mpa) of stress before they rupture. That is, they can deform elastically by less than one per cent before breaking.

When rocks are subjected to a very sudden high stress, far beyond their normal strength, they deform plastically. The limit of elastic deformation that rocks can endure before deforming plastically under such conditions is called the Hugoniot Elastic Limit. It is typically a few tens of kilobars (a few Gpa). This limit is very variable even for the same types of rock but is about an order of magnitude greater than the typical low-pressure strengths of rocks.

Waves in Rocks

Impact cratering and explosions (which have quite similar physics) produce three types of waves in rocks. From lowest to highest pressure, they are:

Elastic Waves

These are the familiar P- and S- waves. Their velocities are given by V(p)=((K + 4G/3)/d)1/2 and V(s)=(G/d)1/2. In these formulas, K is the bulk modulus (a measure of volume compressibility), G is the shear modulus (a measure of shear deformability) and d is the density.

Plastic Waves

Plastic waves move through rocks that are stressed well beyond their normal failure strength. The rocks therefore behave somewhat as fluids, and the velocity of a pressure wave in a fluid is V(f)=(K/d)1/2 (this is just the formula for p-wave velocity in a material with no shear strength, that is, where G=0). Note that this is less than V(p), which equals ((K + 4G/3)/d)1/2. Thus plastic waves are preceded by an elastic precursor wave.

Shock Waves

Shock waves, by definition, travel through materials faster than elastic waves. They simply bulldoze the material by being propelled mechanically by some extremely great force. True shock waves die out very quickly with distance from the source.

Reflected Waves

Wave reflection effects are important in impact mechanics. The easiest case to consider is a pressure or longitudinal wave reflected off a free surface.

Waves at Free Surfaces

The fundamental principle here is that the stress on a free surface must be zero. Thus, if a compressional wave strikes a free surface, and the stress on the surface must always be zero, it follows that the reflected wave must be a rarefaction.

Something very interesting happens at the surface. In a compressional wave the particle motion is toward the direction of travel, thus, toward the free surface. In a rarefaction wave the particle motion is opposite the direction of travel. Since the reflected rarefaction wave is moving away from the surface, the particle motion in the wave is toward the surface. Thus the particle motion in both waves is toward the surface, and the particles at the surface can move at up to twice the particle velocity of the incident wave.

Although the stress on the surface itself must be zero, the stresses behind the surface need not be. The doubled particle velocity can create tensile stresses that rupture the material, causing pieces to spall or fly off at high speed. Spallation plays a
dominant role in excavating impact craters. If you are in a tank that has just been hit by an anti-tank round, in about a millisecond or so spallation will be of intense personal interest to you, though you probably won't survive to describe it.

Waves at Internal Surfaces

If the wave hits a boundary between materials, and the material beyond the boundary has lower elastic wave velocity than the material the wave is traversing, the situation is somewhat like the reflection of a
wave at a free surface. Some of the wave energy continues on as a compressional wave and some is reflected as a rarefaction wave.

If the wave hits a boundary where the material beyond the boundary has higher elastic wave velocity than the material the wave is traversing, most of the wave energy continues on as a compressional wave and some is reflected, also as a compression wave.

Atmospheric Passage

Incoming meteoroids begin to encounter air resistance when the average distance between air molecules (the mean free path) becomes smaller than the size of the meteoroid. For objects a meter across, that altitude is about 110 km, for objects a centimeter across, about 55, and for objects a millimeter across, about 40. Objects a micron across encounter air resistance at about 25 km.

Microscopic particles radiate away frictional heat rapidly enough and slow to a stop so quickly that they survive undamaged; indeed, they are a useful source of data on interplanetary particles. Objects in the millimeter to centimeter range vaporize in flight, becoming visible as "shooting stars". Objects in the meter range may survive entry but be slowed to subsonic speeds by the time they reach the Earth's surface. They fall just like any other rock and may
punch through roofs to dig small craters when they hit. Objects tens of meters in size and larger hit with most or all their kinetic energy intact, resulting in true impact cratering.

Contact and Compression Stage

Imagine a cubic meteor (simplifies the math) 100 meters on a side, striking the Earth at 30 km/sec. Its density is 3000 kg/m3. The mass of the object is thus 3 x 109 kg. It will lose most of its velocity by the time it penetrates a distance equal to its diameter (100 m) and it will take about 100 m/30,000 m/sec = .0033 seconds to do so (actually longer, since it's slowing from 30,000 m/sec to zero, but the most interesting things happen in the brief interval while the velocity is still large). Thus the meteor experiences a deceleration of 30,000 m/sec/.0033 sec = 9 x 106 m/sec2. Since the gravitational acceleration of the Earth is about 10 m/sec2, the meteor experiences about 900,000 g's of deceleration.

Force equals mass times acceleration, so the force the meteor and the Earth are exerting on each other comes to 9 x 106 m/sec2 times 3 x 109 kg = 2.7 x 1016 newtons. The pressure beneath the meteor is simply the force divided by the area, or 2.7 x 1016 divided by 10,000 square meters, or 2.7 x 1012 pascals. Since a bar is 100,000 pascals, the pressure comes to 27 megabars.

Stresses During Contact

Imagine a cubic meteor (simplifies the math) s meters on a side, striking the Earth at v km/sec. Its density is d kg/m3. The mass of the object is thus ds3. It will lose most of its velocity by the time it penetrates a distance equal to its diameter and it will take about s/1000v seconds to do so (actually longer, since it's slowing from high speed to zero, but the most interesting things happen in the brief interval while the velocity is still large). Thus the meteor experiences a deceleration of 1000v/(s/1000v) = (106)(v2)/s.

Force equals mass times acceleration, so the force the meteor and the Earth are exerting on each other comes to ds3(106)(v2)/s = ds2(106)(v2). The pressure beneath the meteor is simply the force divided by the area, or ds2(106)(v2)/s2 = d(106)(v2).
Note that all the size terms have vanished. The pressure is pretty much the
same regardless of the size of the impacting object. What does change is the
radius to which various pressure effects extend.

Although the contact-compression stage lasts milliseconds, two interesting processes occur in this interval. Not only does the impacting object generate a shock wave in the target, but a reflected shock wave propagates upward through the impactor. When it strikes the top free surface it produces a reflected rarefaction wave, resulting in spallation of the impactor.

Also, as the impactor plows into the target, there is typically a wedge-shaped void around the contact,
where the impactor has not yet come into contact with the target. Material being
compressed at the contact can only escape along the axis of the wedge. The
physics is very much like that of an explosive shaped charge. A high-speed jet
of material squirts along the axis of the wedge, a phenomenon known as jetting.

Excavation Stage

During the excavation stage, the impactor plows about its own diameter into
the target. A bowl-shaped cavity, the transient crater, forms. Intensely
compressed material ahead of the impactor is pushed aside and up the sides of
the transient crater, exiting at speeds comparable to the initial velocity of
the impactor. Since the impactor struck with velocity greater than escape
velocity, the fastest ejecta also exceeds escape velocity and can be blasted off
the planet altogether. Meteorites from the Moon and Mars have been found on
Earth. Could there be pieces of Venus or Mercury (as in the film Eight Below)
waiting to be found? There's not a reason why not, but confirming their
identities might be a challenge. There could also be bits of Earth lying about
on other planets. If it's the right kind of rock, say granite or limestone, it
could be very easy to spot.

Modification Stage

After the impactor has stopped (mostly melted by this time, vaporized, or
spalled to bits) and the shock waves have passed, the transient crater begins to
collapse and the modification stage begins:

Craters can take on various forms, depending on their size.
The boundaries between different crater forms are inversely proportional to the
planet's gravity, so the transition between simple craters and central peak
craters might occur at 10 km diameter on Earth and Venus, 30 km on Mars and 60
km on the Moon.